ThA01.1
43rd IEEE Conference on Decision and Control
December 14-17, 2004
Atlantis, Paradise Island, Bahamas
Optimal Allocation of a Divisible Good to
Strategic Buyers
Sujay Sanghavi and Bruce Hajek
Electrical Engineering, University of Illinois at Urbana Champaign
[email protected], [email protected]
Abstract— We address the problem of allocating a divisible
resource to buyers who value the quantity they receive, but
strategize to maximize their net payoff (value minus payment).
An allocation mechanism is used to allocate the resource
based on bids declared by the buyers. The bids are equal
to the payments, and the buyers are assumed to be in Nash
equilibrium. For two buyers such an allocation mechanism
is found that guarantees that the aggregate value is always
greater than 87 of the maximum possible, and it is shown that
no other mechanism achieves a larger ratio. For a general
finite number of buyers an allocation mechanism is given
and an expression is given for its worst case efficiency. For
three buyers the expression evaluates to 0.8737, for four
buyers to 0.8735 and numerical computations suggest that the
numerical value does not decrease when the number of buyers
is increased beyond four. A potential application of this work
is the allocation of communication bandwidth on a single link.
I. I NTRODUCTION
Players in non-cooperative games try to maximize their
own payoff functions. If such a game has a designer with
preferences on the outcomes, it may be possible for the
designer to decide on strategy spaces and the corresponding
outcomes (i.e. the mechanism) so that the players’ strategic
behavior will not lead to an outcome that is far from
desirable.
Consider the game involving the allocation of a single
unit of a divisible resource to competing buyers each of
whom has to make a payment in compensation. Each
buyer obtains a certain amount of value from allocations
of the resource made to it, but strategizes to maximize
his net payoff - the value minus payment. An allocation
of the resource that maximizes the aggregate value (given
the buyer value functions) is said to be efficient, other
allocations with lower aggregate value are inefficient.
Nash equilibria are fundamental in the study of games of
this nature. That inefficient allocations may occur at Nash
equilibria is well known in the economics literature [9].
The question addressed in this paper is: how to design
an allocation mechanism that results in the worst-case
inefficiency of a Nash equilibrium being as low as possible
?
0-7803-8682-5/04/$20.00 ©2004 IEEE
Recent research contains many examples of efforts to
quantify the inefficiency of Nash equilibria in games related to resource allocation. Koutsoupias and Papadimitriou
[5] and Roughgarden and Tardos [4] quantify worst-case
inefficiencies of Nash equilibria in routing games. Johari
and Tsitsiklis [1] quantify the inefficiency in allocating a
divisible good with a uniform price. All of these papers use
the ratio of the welfare at Nash equilibrium to the welfare
at the social optimum, thus evaluating fractional efficiency.
Auctions of divisible goods have also received much
attention. Besides [1] mentioned above, Maheswaran and
Başar [7] and Gopalkrishnan and Hajek [2] also deal with
allocation of a single divisible good. Kelly [6] shows that
if the buyers are not strategic but price-taking, allocation of
a divisible good can be made in a socially optimum way.
In the economics literature this model appears in Back and
Zender [8] who advocate using discriminatory pricing for
the sale of treasury bonds.
For two buyers this paper finds the mechanism that has
the lowest worst-case fractional inefficiency for allocating
a divisible good when the bids that the buyers place are
the payments they will make. The mechanism ensures that
any Nash equilibrium (for any set of buyer preferences) will
th
be at least 78 as efficient as the optimal allocation. The
extension to n buyers has not been completely evaluated,
but numerical computations suggest that the worst possible
case (with any number of buyers) has a fractional efficiency
of 0.8735. Other characteristics like uniqueness of equilibria
are also investigated.
II. T HE S ETUP
Consider the situation where one unit of a divisible good
is to be split up among n buyers. Let w = [w1 , . . . , wn ] be
the (non-negative) payments that the buyers make and let
x = [x1 , . . . , xn ] be the quantities they are allocated as a
result. The allocation is made according to a pre-specified
allocation mechanism τ , so that given the payments w the
allocation to buyer i is given by xi = τi (w).
An allocation rule τ is said to be a valid allocation
mechanism if it has the following properties:
≥ 0 and i τi (w) = 1 for all
A1 It is an allocation: τi values of w such that i wi > 0. Also, τi (0, w−i ) =
2748
0 for all w−i i.e. a zero bid wi = 0 will get zero
allocation (even if all bids zero).
A2 It is smooth: τi (wi , w−i ) is differentiable, increasing
and concave in wi for all w−i , except in the case
∂τi
(w)
when w−i = 0. Also for each i, τi (w) and ∂w
i
are continuous in the vector of payments w over the
set Rn+ − {0}.
A3 It is symmetric in the buyer indices: τi (w) =
τσ(i) (σ(w)) for all permutations σ of the indices
i = 1, . . . , n
A4 It is scale free: for all real γ > 0 and 0 ≤ i ≤ n,
τi (γw) = τi (w). 1
One example of a mechanism in the class above is the
proportional allocation (or uniform price mechanism) that
divides the good so that each buyer gets a quantity proportional to the payment it made.
The buyers each have a value function Ui (xi ) for the
amount of the good they are allocated. A value function is
said to be valid if it is differentiable, concave and strictly
increasing, with Ui (0) = 0. This paper deals only with valid
value functions.
The aggregate value (or social welfare) of an allocation
x is the sum of the individual values. For a set of value
functions, an allocation x∗ is efficient if it has the largest
total value:
Ui (x∗i ) ≥
Ui (xi ) ∀ x
i
i
Whether an allocation is efficient depends only on the
functions Ui and not on τ . The efficiency of any allocation
x is defined to be the fraction
U (x )
i i ∗i
U
i i (xi )
By definition the efficiency lies between 0 and 1.
Given the payments w, the profit Pi of each buyer is the
value derived from the allocation minus the payment made:
Pi (wi , w−i ) = Ui (τi (w)) − wi
Note that Pi is concave in wi for every fixed value of the
other amounts w−i .
w
i > 0 for at least two buyers i and
∂ τi
= 0 if w
i > 0
(w)
−1
Ui (τi (w))
∂wi
≤ 0 if w
i = 0
(1)
for all i = 1, . . . , n.
It can be shown that Nash equilibria exist with the
assumptions stated. The way to do this is to consider, for
the given set of value functions Ui , the -game where
each of the buyers are forced to bid at least . Also,
the payments would never exceed the value obtained from
having the entire unit of the resource, so we can assume
that wi ≤ Ui (1). Then, by Theorem 1 in Rosen [3] there
exists an NEP w
for the -game. A sequence n → 0 will
have a sequence of corresponding equilibria w
n . Consider
a convergent subsequence in the sequence of equilibria,
and let w
be its limit point. Then it can be shown that w
will satisfy the necessary and sufficient conditions as stated
above. 2
Given a set of value functions {Ui } and valid allocation
mechanism τ , let N (τ, {Ui }) be the set of Nash equilibria
and let x∗ be an efficient allocation. Then, the worst case
efficiency when the good is allocated according to τ is given
by
i Ui (τi (w))
inf inf ∗)
e
U
(x
{Ui } w∈N
i i i
The worst case is taken over a very broad set of buyer
preferences and numbers, and indeed many allocation mechanisms are likely to have zero worst-case efficiency. Stated
in these terms, [1] proves that the worst case efficiency of
the proportional allocation mechanism is 34 .
Our objective is to find the allocation mechanism with
the highest worst-case efficiency, in the class of valid
mechanisms (i.e. those that satisfy assumptions A1-4). We
also want to find the value of this best possible worst case
efficiency:
i Ui (τi (w))
(2)
sup inf inf ∗)
e
w∈N
U
(x
{U
}
i
τ
i i i
III. R EDUCTION TO L INEAR VALUE F UNCTIONS
to be an Nash equilibrium, i.e. for w
∈
For a vector w
N (τ, {Ui }) the necessary and sufficient conditions are that
Following [1], which considered only the proportional
allocation mechanism, it is shown in this section that for the
purposes of evaluating the worst case performance of any
valid allocation it is sufficient to assume that the buyers have
linear value functions. This is crucial to the analysis that
follows, for two reasons: it simplifies the space over which
the infimum has to be taken, and the efficient allocation has
the simple form of giving all to the buyer with the highest
slope. Note that the results in this section do not use the
scale-free assumption A4 and have no restrictions on the
number of buyers except that it be finite.
1 Stated differently, the mechanism is independent of the units in which
the payments are made.
2 If assumption A2 does not hold it is possible to construct examples
where there are no Nash equilibria.
The buyers play a non-zero-sum non-cooperative game
with strategy variables wi and payoff (or reward) functions
is said to be a Nash
Pi respectively. A vector of payments w
equilibrium if individual deviations (changes in payment
made) cannot help a buyer:
−i ) ≥ Pi (wi , w
i ) ∀ i, wi ≥ 0
Pi (w
i , w
2749
Lemma 1 is reproduced from [1], where it appears as
Lemma 4. The reader is referred to the original paper for a
proof.
Lemma 1 (Johari and Tsitsiklis [1]): For any value
functions {Ui } if an efficient allocation is x∗ and x is any
other allocation, the following inequality holds:
U (x )
i Ui (xi )xi
i i ∗i ≥
maxi Ui (xi )
i Ui (xi )
As in [1], Lemma 1 implies that considering only linear
value functions for the buyers is sufficient.
Proposition 1: For every τ satisfying A1-3, valid value
∈ N (τ, {Ui }) there exist linear value
functions {Ui } and w
i }) =
∈ N (τ, {U
functions Ui (xi ) = αi xi such that w
N (τ, α) and
Ui (τi (w))
i
i αi τi (w)
≥
(3)
∗)
U
(x
max
α
i i
i i i
where x∗ is an efficient allocation for {Ui }.
satisfies the
∈ N (τ, {Ui }), the vector w
Proof: Since w
conditions (1). Consider linear value functions defined by
i (xi ) = αi xi = Ui (τi (w))x
i
U
∈ N (τ, α) by checking that the
It is easy to see that w
conditions (1) hold for the new linear value functions αi xi .
Equation (3) is exactly the same as the statement of Lemma
and writing Ui (xi ) = αi in the RHS. 1 with xi = τi (w)
Note that the LHS of (3) is the efficiency of a Nash
equilibrium for τ, {Ui } and the RHS is the efficiency of
the same Nash equilibrium for τ, α (since for linear value
functions an efficient allocation gives all of the good to the
buyer with the biggest slope). Thus Proposition 1 says that
the efficiency of a particular Nash equilibrium pair is lowest
when the corresponding value functions are linear. Thus, for
the infimum in the objective function (2) it is sufficient to
consider linear value functions for both the users.
If αi xi are the value functions of the buyers and N (τ, α)
is the set of Nash equilibria for these under allocation mechanism τ then the objective function (2) can be rewritten as
i αi τi (w)
sup inf
inf
(4)
α w∈N
e
(τ,α) maxi αi
τ
The rest of the analysis in this paper deals with the
objective function given above.
IV. T WO B UYERS
Consider first the case of there being only two buyers.
Since assumption A3 asks for invariance with respect to
permutation of indices, it is sufficient for the two-buyer case
to specify a mechanism by the amount it allocates to the
higher buyer (who makes the larger payment) and to the
lower buyer.
Let wl be the lower payment made and wh be the higher
payment, i.e. wl ≤ wh and let τ ∗ (wl , wh ) be the allocation
mechanism given by
wl
wl
and τh∗ = 1 −
(5)
τl∗ =
2wh
2wh
where τl∗ is the allocation to the lower buyer and τh∗ is
to the higher buyer. In this section we will prove that this
is indeed the optimal valid mechanism: it has the highest
worst-case efficiency.
The price (payment made for unit quantity received) each
buyer pays is different, making this a discriminatory price
mechanism with the prices determined by the willingness of
each buyer to pay. The mechanism has a “volume discount”:
a higher buyer pays a lower price, and so has an incentive
to bid high and - as a result - get a higher allocation. This
discount partially offsets the effect that a high buyer tends
to bid up the price and work against himself, causing a
strategic high buyer to buy less quantity than is efficient.
The following lemma is easily verified, and we omit the
proof.
Lemma 2: τ ∗ is a valid allocation mechanism.
The theorem that follows shows that this is the optimal
valid two-buyer mechanism. First though we need a proposition:
Proposition 2: For any two-buyer valid allocation mechanism τ there exists a function φ : [0, 1] → [0, 1] such that
wl
wl
and τh (wl , wh ) = 1 − φ
τl (wl , wh ) = φ
wh
wh
whenever wh > 0. We say that the allocation τ is based on
φ. The function φ further satisfies the following properties:
B1 0 ≤ φ(v) ≤ 1 for all v ∈ [0, 1], with φ(0) = 0 and
φ(1) = 21
B2 φ(v) is differentiable, increasing and concave in v for
all v ∈ [0, 1].
Proof: Given a valid allocation τ , define φ(v) = τl (v, 1),
i.e. the allocation to the lower buyer when the lower
payment is v and the higher is 1. Then, it is easy to see
that when wh > 0,
wl
wl
,1
= φ
τl (wl , wh ) = τl
wh
wh
where the first equality follows from the scale-free assumption A4 with γ = w1h . This and A1 immediately imply that
τh (wl , wh ) = 1 − φ( wwhl ) and that 0 ≤ φ(v) ≤ 1. Also
by A1, when the lower payment is 0 the lower allocation
is 0, and this gives us φ(0) = 0. When both payments
are equal and non-zero, say wl = wh = w > 0, the
2750
symmetry assumption A3 implies that τl (w, w) = 12 and
hence φ(1) = 12 . Thus φ satisfies B1. The assumption A2
that τl be increasing, concave and differentiable in wl when
wh > 0 implies that B2 holds for φ. Thus the proposition
is proved.
Notice that the converse does not hold: not all functions
φ satisfying B1, B2 correspond to valid allocation mechanisms, since B1, B2 do not address the issue of concavity
and differentiability in one of the payments w1 at the point
of the other payment w2 , i.e. in the region where w1 < w2
becomes w1 > w2 . Maximizing the worst case efficiency
over the space of functions satisfying B1, B2 thus yields an
upper bound on the efficiency of valid allocations:
i αi τi (w)
sup inf
inf
α w∈N
e
(τ,α) maxi αi
τ
i αi φi (w)
≤ sup inf
inf
α
e
max
α
w∈N
(φ,α)
i
i
φ
where φi is the allocation to buyer i from an allocation
based on function φ. The following theorem evaluates the
right hand side for two users.
Theorem 1: When two buyers are present, the worst case
efficiency of τ ∗ is 78 , and no valid mechanism can achieve
a higher ratio.
Proof: We will first prove that
and then show achievability.
7
8
represents an upper bound,
By the reasoning in Section III, for the purposes of
worst case analysis it is sufficient to consider linear value
functions with slopes αl and αh for the two users, both
being strictly positive. In light of Proposition 2 above we
h ) ∈
will consider functions φ satisfying B1-2. If (w
l , w
N (φ, α), then the Nash equilibrium conditions of (1) give
l
l
w
l w
1 w
1
1
φ
and
φ
=
=
2
w
h
w
h
αl
w
h
w
h
αh
The above equations imply that for a given pair of linear
value functions there will be unique Nash equilibrium
h ) and combining the two equations above gives
(w
l , w
αl
w
el
=
αh
w
eh . The efficiency of the unique Nash equilibrium
for the given slopes is thus
el
w
el
+
α
1
−
φ
αl φ ww
h
eh
w
eh
Eα =
αh
Minimizing this over all α such that αl ≤ αh will give the
worst case performance of φ, and maximizing this over all
φ will give the upper bound. Denoting v = ααhl , the upper
bound is given by
sup min vφ(v) + 1 − φ(v)
φ
achieves the sup above, and the upper bound can be
evaluated:
v2
v
7
+1−
=
(6)
min
0≤v≤1 2
2
8
where the minimum is achieved at v = 12 . Thus 87 represents
an upper bound on the worst case performance of valid
mechanisms.
For achievability, note that τ ∗ is the mechanism based on
φ (v) = v2 (in the sense of Proposition 2), and Lemma 2
guarantees that it is valid. All that is needed now is a direct
verification of the worst case performance.
∗
As before, if αl and αh are the slopes of the value
functions of the buyers, then there will be a unique Nash
equilibrium (w
l , w
h ) and furthermore we have that ααhl =
w
el
w
eh . Thus the worst-case efficiency of a Nash equilibrium
for τ ∗ is given by
αl
αl
αl
min
+1−
αl ≤αh αh
2αh
2αh
Denoting v = ααhl makes this exactly the same as the LHS of
equation (6), and hence the worst case efficiency is indeed
7
8.
This worst case occurs when the slope of the value
function of one buyer is half that of the other.
Unique Equilibria in τ ∗
While the existence of Nash equilibria is guaranteed,
uniqueness is not: it is possible that more than one Nash
equilibria exist for a given set of value functions of the
buyers and an allocation mechanism. There are many ways
in which the notion of an equilibrium in a game can be
refined to “choose” one of them. It would however still be
of interest to see if given an allocation mechanism τ there
is always a unique Nash equilibrium for any set of buyer
value functions.
The proof of Theorem 1 indicates that allocating according to τ ∗ results in a unique Nash equilibrium when the
buyers have linear value functions. The following theorem
proves that this is the case for any pair of buyer value
functions.
Theorem 2: For the case when there are only two buyers,
the allocation mechanism τ ∗ results in a unique Nash
equilibrium for any pair of valid buyer value functions
U1 , U2 .
Proof: Let 1 and 2 be the two buyers, and N (τ ∗ , U1 , U2 )
be the set of Nash equilibria when allocation is done
according to τ ∗ . Define two types of Nash equilibria:
0≤v≤1
The function φ∗ (v) = v2 satisfies B1-2 and has φ(v) ≤
φ∗ (v) for any other φ that also satisfies B1-2. Thus it
Type I:
Type II:
2 ) ∈ N (τ ∗ , U1 , U2 ) with w
1 ≤ w
2
(w
1 , w
(w
1 , w
2 ) ∈ N (τ ∗ , U1 , U2 ) with w
1 ≥ w
2
Note that an equilibrium will be of both types iff
2751
w
1 = w
2 . Any Nash equilibrium has to satisfy the
condition given by (1). For τ ∗ allocation and Type I
equilibria this can be rewritten as w
1 > 0, w
2 > 0 and
1
1 w
w
1 w
1
U
U 1−
=
= w
2
2 1 2w
2
2w
2 2
2w
2
2 (x) as follows:
From U2 (x) define U
2 (0) = 0 and U
(x) = U (x)(1 − x)
U
2
2
2 is strictly concave for all valid U2 . Also, if
Note that U
we define x
= 2wewe12 then we have that
x
= arg max
0≤x≤1
U1 (x) + U2 (1 − x)
2
The above function is strictly concave in x, hence there
exists a unique x
maximizing it. For there to exist Type I
equilibria the maximum above should be achieved with x
≤
1
:
if
this
is
not
the
case
there
will
be
no
Type
I
equilibrium.
2
This and the fact that w
2 = 12 U1 2wewe12 means that there
can be at most one Type I equilibrium. By similar reasoning
there can be at most one Type II equilibrium. Also, at least
one Nash equilibrium point exists. Thus all we need to prove
is that if there is an equilibrium of each type, then these
two equilibria are the same point.
1 from U1 in the same way that U
2
To do this, define U
was defined using U2 , and let
1 (1 − y) + U2 (y)
y = arg max U
0≤y≤1
2
Suppose equilibria of both types exist. Then, from the
definitions of x
and y we have that
U1 (
x)
2
= x
U2 (1 − x
)
(7)
y)
U2 (
2
(8)
y U1 (1 − y) =
Also, we should have that x
≤
1
2
and y ≤ 12 . This gives us
„ «
„ «
x)
U (e
1
1 1
≤
U1
≤ 1
2
2
2
2
„ «
„ «
y)
U (e
1
1 1
e) ≤ x
e U2
x
e U2 (1 − x
≤
U2
≤ 2
2
2
2
2
ye U1 (1 − ye) ≤ ye U1
V. n B UYERS
The optimal valid mechanism for two buyers has a natural
extension to the case when there are n buyers. Specifically,
we saw that the optimal two-buyer mechanism was the one
that gave a higher buyer a lower price as compared to a
lower buyer, and these prices were determined by the total
amounts each buyer was willing to pay.
In this section we will omit the proofs, since they use
essentially the same ideas as in the two-buyer case but are
more technical.
Given a payment vector w let wmax denote the maximum
payment, and consider the allocation mechanism τ ∗ that
allocates to each buyer i according to
τi∗
1
s=0 j=i
wj
1−s
wmax
ds
Note that for n = 2 it simplifies to the two-buyer τ ∗ of
the previous section. It is easy to see that this allocation is
symmetric and scale-free, i.e. that it satisfies A3 and A4.
Verifying A1 and A2 is technical but straightforward, we
do not include the proof here but state the result below.
Lemma 3: The mechanism τ ∗ as defined above is valid,
i.e. it satisfies assumptions A1-4.
Following the method of Theorem 2 above, we can derive
an expression for the worst-case efficiency of τ ∗ . Unfortunately we have not been able to evaluate it analytically for
general n:
Theorem 3: The worst case efficiency En of τ ∗ is given
by
n−1
n−1
1 vi
(1 − svi ) ds
1+
En = minv∈[0,1]n−1
0
i=1
n−1
n−1 i=1
−
(1 − vi )
vi
(10)
Thus, there is a unique equilibrium for the two-player game
when allocation is done according to τ ∗ .
if wmax > 0 and τi∗ = 0 for all i if wmax = 0. The value
of the empty product is taken to be 1.
(9)
Together (7)-(10) give us that x
= y = 12 , which for given
Ui means that there is only one NEP given by
1
1
1
1
w
1 = w
2 = U1
= U2
2
2
2
2
wi
=
wmax
i=1
(11)
i=1
It can be shown analytically that for n = 2 the above
gives a value of 87 , and for n = 3 it gives 0.8737. This
compares favorably with the proportional allocation, which
√
was shown in [1] to have a worst case efficiency of 2( 2 −
1) ≈ 0.8284 when there are two buyers.
The minimum for n ≥ 4 is still unsolved analytically, but
numerical computations suggest (counter-intuitively) that
the worst case over all n occurs when n = 4, i.e. there
are only 4 users with non-zero payments. The value then is
0.8735. We state this as a conjecture:
The above technique shows the uniqueness of Nash equilibria by formulating an equivalent optimization problem.
Similar methodologies have been used before in [2] and
also in [1].
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Conjecture 1: With efficiency En as defined above, the
worst case efficiency occurs when n = 4:
inf En = E4 = 0.8735
n≥2
If true, the above conjecture means that τ ∗ compares
quite favorably to the proportional allocation: not only is the
worst case ratio for τ ∗ higher than the 34 for the proportional
allocation, there is also no degradation as the number of
buyers grows.
An interesting property of τ ∗ is that it is the unique
“linear in the lower bids” mechanism:
Lemma 4: If τ is an allocation mechanism satisfying
assumptions A1-3 and also has the following property
L Linearity: τi (w) is linear in wi in the region 0 ≤
wi ≤ maxj=i wj , when maxj=i wj > 0,
Another interesting (and obvious) feature of the general
allocation τ ∗ with n buyers is its hierarchical property: if m
of the n buyers submit a payment of 0 (or are “phantom”),
the resulting allocation would be the same as it would have
been if the n−m “serious” buyers were allocated quantities
according to the τ ∗ with n − m buyers.
Extending the above sort of analysis to multiple interrelated goods (like e.g. capacities on links in a network) is
also of interest. Specifically, it would be interesting to see
if the worst case buyer value functions turn out to be linear
in that case as well.
Besides this, another question in the single-good scenario
is one of revenue maximization for the seller. A similar
fractional formulation for the problem of either worst-case
or best-case revenue loss due to strategizing can also be
made, and allocation mechanisms for minimizing this can
be found. Again, it would be of interest to see if the extremal
value functions in this scenario are linear.
then τ = τ ∗ .
It is obvious from (11) that property L holds for τ ∗ .
Property L has an interesting feature when the value functions of the lower buyers are linear: at Nash equilibrium the
payment of each lower buyers equals the it obtains from the
allocation.
VII. ACKNOWLEDGMENT
The authors are grateful to Prof. Steven R. Williams for
his comments and encouragement.
VI. C ONCLUSION AND E XTENSIONS
This paper proposes a mechanism for allocating a good
to buyers based only on the payments they make, so that the
loss in value created due to strategizing is minimized. The
mechanism gives the lowest price to the highest buyer, thus
giving an incentive for buyers who value the good more to
bid higher. This results in an increase in efficiency.
It is however not yet a complete piece of work. The
most immediate task is of course to prove (or disprove)
the optimality of τ ∗ for n ≥ 3, and evaluate the worst case
value given by (11). A second task is to determine whether
Theorem 1 is still true if assumption A4 is dropped. A third
task is to see if the τ ∗ allocation rule with n buyers has a
unique Nash equilibrium for general buyer value functions,
possibly along the lines of the techniques used for the twobuyer case.
Notice that τ ∗ is a discriminatory price auction in which a
buyer with a larger payment pays a strictly lower price than
one with a strictly lower payment. This would suggest that
the mechanism is immune to strategic splitting, wherein one
buyer enters the game as multiple buyers who collaborate.
The pricing suggests that it would be in its best interests
to avoid splitting, as for the same amount of total payment
made it would receive the highest quantity of resource if
it made a single large payment. On the other hand though,
this reasoning suggests that buyers will stand to gain by
cooperating. These notions have to be formalized.
2753
R EFERENCES
[1] R. Johari and J. Tsitsiklis, “Efficiency loss in a network resource
allocation game,” Mathematics of Operations Research 29 (3) pp.
407-435
[2] B. Hajek and G. Gopalakrishnan, “Do greedy autonomous systems
make for a sensible Internet?” Presentation at the Conference on
Stochastic Networks, Stanford University, June 2002.
[3] J. B. Rosen, “Existence and uniqueness of equilibrium points for
concave N-person games,” Econometrica, Vol. 33, Issue 3, pp. 52034, July 1965.
[4] T. Roughgarden and E. Tardos, “How bad is selfish routing?” Journal
of the ACM vol. 49, no. 2, pp. 236-259, 2002
[5] E. Koutsoupias and C. Papadimitriou, “Worst case equilibria,” Proceedings of the 16th Annual ACM Symposium on Theoretical Aspects
of Computer Science, pp. 404-413, 1999.
[6] F. Kelly, “Charging and rate control for elastic traffic,” European
Transactions on Telecommunications vol. 8, pp. 33-37, 1997.
[7] R. Maheswaran and T. Başar, “Nash equilibrium and decentralized
negotiation in auctioning divisible resources,” J. Group Decision and
Negotiation (GDN), 13(2), May 2003.
[8] K. Back and J. Zender, “Auctions of divisible goods: on the rationale
of the treasury experiment,” The review of Financial Studies vol. 6,
no. 4, pp. 733-64, 1993
[9] P. Dubey, “Inefficiency of Nash Equilibria,” Mathematics of Operations Research vol 11, pp. 1-8, 1986